Multi-particle composites in density-imbalanced quantum fluids

Thanks to recent advances in experimental techniques of dealing with cold gases, it is now feasible to engineer one-dimensional (1D) quantum fluids by confining atoms in cigar-shaped traps with tight radial confinement [1]. By devising an appropriate optical lattice it is also possible to construct a weakly coupled array of such 1D "tubes," thus allowing one to study the dimensional crossover from 1D to three dimensions. A number of ongoing and planned experiments deals with two-component mixtures atoms of either statistics, i.e. Fermi-Fermi (FF), Bose-Bose (BB) or Bose-Fermi (BF) mixtures [2]. Most of recent theoretical work dealt with equal-density mixtures, where a rich phase diagram containing both gapped and gapless phases was found [3,4,5]. For mixtures with unequal densities the ground state is generally found to be a two-component Luttinger liquid [3,6,7]. For attractive FF mixtures, superconducting correlations dominate, thus making the ground state a 1D analog of the long-elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, as confirmed both by Bethe Ansatz calculations for integrable models [8] and numerical simulations [9,10]. The case of unequal mass mixtures-where integrable microscopic models are not available-has been studied analytically by means of effective field theory [3,7], and numerically by Monte Carlo [11] and time-evolving block decimation (TEBD) [10] methods. A common result which emerges is that for strong enough mass asymmetry and/or strong enough attraction the system collapses, while for moderate mass asymmetry and non-zero density imbalance the ground state is again a gapless two-component Luttinger liquid with an FFLO-type algebraic order.

In this Letter we study a generic two-component 1D mixture with density imbalance within the harmonic fluid approach (“bosonization”). We reveal a generic mechanism which, for a certain relation between the densities, opens a gap in the excitation spectrum and completely destroys superconducting correlations. We concentrate on the properties of the FF mixtures, but our predictions are applicable to BF and BB mixtures with minor modifications. Our findings might also be relevant to spin ladder materials with non-equivalent chains in high magnetic fields. We further corroborate our predictions by DMRG simulations [12] of a Hubbard model with hopping asymmetry.

Consider the mixture of two sorts of fermionic atoms, which we label by a pseudo-spin index σ =↑, ↓. In the bosonization approach we introduce for each species a pair of scalar fields φ σ (x) and θ σ (x) which vary slowly on the scale of n -1 σ , where n σ are the average densities [13]. Using the Haldane construction we write for the field operators

s e is(πnσ x-φσ) e -iθσ where the summation over s runs over odd integers s. [14] For the density operator, nσ , this leads to :

One of the advantages of the Haldane representation (1) is that an effective low-energy Hamiltonian can be written solely in terms of φ σ and Π σ [13]. In the noninteracting case it is given by H free = H 0 (φ ↑ ) + H 0 (φ ↓ ), where :

where v σ are Fermi velocities and K σ = 1 the so-called Luttinger parameters equal to one in the free case. In presence of density-density interactions, dxdx ′ U σσ ′ (xx ′ )n σ (x)n ′ σ (x ′ ), Eq. ( 2) is modified in several ways. First of all, the s = 0 terms of Eq. (1) give rise to an acoustic coupling :

where g is a forward scattering constant for the interspin interactions. More importantly, higher harmonics of Eq.

(1) generate the terms of the form :

Here G s,s ′ and Gs,s ′ are (non-universal) amplitudes, and k σ F = πn σ are Fermi momenta. Since the separation of fast and slow variables is inherent in the bosonization treatment, one has to discard in (4) the terms which oscillate on the lengthscale ∼ k -1 F . Strictly speaking, Eq. ( 3) is only perturbative in g. On the opposite, Eq. ( 2) is assumed to retain its functional form even in presence of generic same-spin density-density interaction, with both velocities and Luttinger liquid parameters renormalized by interaction terms beyond Eq. ( 3) and various irrelevant operators, e.g. band curvature [13]. On a phenomenological level, we can assume Eq. ( 2) (where, in general, K σ = 1) as coming from an underlying microscopic model, with Eqs. ( 3) and ( 4) regarded as perturbations.

Equation ( 4) suggests considering generalized commensurabilities of the form

where p and q are relatively prime integers. Notice that this condition does not imply the presence of a lattice : we only require the densities to be commensurate with each other. Eq. ( 5) selects from Eq. ( 4) the terms with s/s ′ = p/q, and the Hamiltonian (4) reduces to

where we only keep the lowest order term, since the scaling dimension of the operator cos sφ is s 2 .

We now assume that the densities are commensurate via (5), and analyze the model 2)-( 3) and (6). Since in general this model is not exactly solvable, the nature of the phases can, in p

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